I know of no such law. Apparently someone from the IL State Police has been giving erroneous information to out of state companies selling ammunition to IL residents. In my search of the IL gun laws, there is nothing that disallows ammo shipments to IL. At least one IL based online retailer, has told me that the state police told them that they couldn't sell to IL residents through UPS, etc., that they had to see an IL resident's FOID card in person, which may be true for IL residents purchasing from an IL retailer, but buying from out of state is not prohibited by federal law, and I see no law preventing it in the IL laws.
Some retailers, AIM, Midway (as of early this year), and a few others that I can't think of right now, have at least temporarily been intimidated by IL law enforcement or the Attorney General's office. A member here, don't remember who, sent a letter to Midway and got a response, somewhat explaining the situation without going into detail. I can't find the thread right now because I can't use the search feature.